Related to this question. The Science world, of course, has experienced and discarded gold-standard and is using paper currency and floating exchange rates. The magic world, on the other hand, uses gold, silver, and bronze coins. The problem is that the two worlds are connected by magical portals located near the Karman Line (100 km) above sea level. Only through rockets or SSTO spaceplanes can actual goods be moved from the science world to the magical world. If people are trying to go to the other world through magical means, they could not carry anything with them other than a magic-tech smartphone. So carrying goods over world borders is hard, but not impossible. The need for the science world to research magic and use magic to assist scientific research and manufacturing is huge, as well as the need for the magical world to purchase advanced technology from the science world. What kind of exchange rate would be suitable for this world, given that trying to get the other world to accept unequal treaties through military means across the portal is next to impossible?
I think you're putting the cart before the horse
The exchange rate from the magic world to the scientific world is trivial. It's impossible to believe your scientific world wouldn't have something akin to a commodities exchange where the value of gold, silver, tin, and copper (together making bronze) in the scientific world is set and, therefore, well known.
So, the value of your gold, silver, and bronze coins is already known in the scientific world, even if it's pure opinion what that value would be if we, here at Worldbuilding, were to try to put a number to it.
The problem is what a piece of paper is worth to your magic world. How much paper they'd receive (or, more accurately, what the expected exchange value for goods in the scientific world...) is also already known. Based on the commodity exchanges in the scientific world, a gold piece would be worth a known number of "dollars" that that number of "dollars" worth of worth would be given to the magic world.
But what's the value to the magic world?
You haven't explained anything that can be used to judge exchange rates. What the cost of building a house in the magic world? OK! Once we know how many gold pieces a new house costs, we can compare that to the number of "dollars" those same gold pieces are worth, and then compare the number of "dollars" received to how many "dollars" are required to build a similar home in the scientific world.
The result? Now you know exactly what the exchange rate is, because you know the compensating reference (aka, the ratio) between the expected value of the two homes compared to the known value of the gold based on the commodity exchange.
Have I answered your question?
I doubt it — because you haven't told us anything about what anything costs in either of the two worlds. I almost voted to close the question because of that, but I decided you could benefit from where the basic references for establishing a rate of exchange can come from.
So, go and invent the commodity exchange on the scientific world and go and figure out what two equivalent homes would be in both worlds, then decide what their values will be. Ratio of the difference in home prices vs. the commodity value of gold sets your exchange rate.
A bit of a cop-out maybe but the Science world has a material cost listed for gold, silver, and copper. They can use that.
Another option would be bartering. Ill sell you this spell-book of mine that details how i do my magic. In return i want that laptop-computer of yours.
Both may be used. In the magical world they barter. Then when they go back they use the price of gold to calculate the numbers for their accounting sheets.
I don't think exchange rates are the right way to think about this problem.
When you say "exchange rate," I think of exchanging currencies. At the time I write this, the exchange rate between euro and US dollars is about 1:1.01. We get at that number by speculation- people in the know who can choose between dealing in euro and US dollars on average at the moment figure they get about as much purchasing power from 1€ as from $1.01. It's a bit more complicated than that, as people speculate about the future exchange rate, but that's what it boils down to.
It's pretty easy to exchange currencies in our modern world though, your setting looks a lot more like transporting goods long distances. Although gold/silver/copper pieces are used as currency in the magical world, in the science world they're more likely to be novelty items or objects of study. Currency exchange rates are already mostly arbitrary, but that's even more true in this case since these pieces wouldn't really be perceived as currency. This is even more true in the magical world, since gold and silver at least have some innate value whereas paper fiat currency does not.
All of which is to say, trade between the worlds probably wouldn't be done in terms of money, but in terms of what value each can provide to the other. Like in the other answer, they'd be likely to barter directly for whatever objects would be useful to research, taking into account the expense of actually transporting the goods up to and from the portals.
Same as with two countries
The exchange rate between currency of any two countries changes day by day and is determined by bartering and supply/demand.
Each country decides a rate they are willing to sell their currency, but has no control over the rate they can buy the other currency.
Each country wants to buy the other's currency because it can be spent over there. The conversion rate between Magic World and Science World is no different.
It will make a difference if the governments have to deal with each other directly, or whether the citizens can move back and forth, bringing currency and goods to buy and sell in the other place.
In the latter case the currencies will go on the stock market and the countries will be forced to exchange at stock market rates. There is no reason to buy goods in MGD when I can get a better deal by first converting the MGD to SD on the stock market, and then buy the goods in SD.
This is complicated because MGD are valuable not only to buy things, but also as gold in itself. So the price of MGD in Dollarydoos will never go below the price of the gold itself, minus processing costs.
In the former case the exchange rate is whatever the two countries agree to. There is no greater arbiter than capitalism. Science World can at any point decide the conversion rate is 1SD = 1,000,000 MGD. Of course this will result in Magic World refusing to trade. But maybe threatening to pull out of the trade deal is a good bartering tactic.
The exchange rate would not get "set up", it would develop naturally based on supply and demand. Just like it happens with currencies on Earth.
First, let's consider a scenario which does not require any currency exchange. Let's imagine you are a trader on TechWorld and you want to make a profit by trading with MagiWorld. You know that there are some TechGoods which are abundant on TechWorld but very sought out after on MagiWorld. Similarly there are MagiGoods which are easy to get on MagiWorld, but very popular on TechWorld. What do you do?
- You use your TechMoney to buy TechGoods
- You travel to MagiWorld
- You sell your TechGoods for MagiMoney
- You immediately spend that MagiMoney on MagiGoods
- You travel back to TechWorld
- You sell your MagiGoods for TechMoney. And if you were a clever trader, you end up with more TechMoney than you started out with.
In that case you don't need an exchange rate, because you only trade in local currency for local prices.
OK, but what if you only want to import something but don't want to export anything? You can't easily obtain MagiMoney, because you don't have anything to sell on MagiWorld. So what you need to do is find someone on TechWorld who has the opposite problem: They are exporting but not importing anything, so they have a ton of MagiMoney which nobody on TechWorld accepts. So you go to them and make a deal with them to buy their MagiMoney for your TechMoney, so you can travel to MagiWorld and go shopping.
The exchange rate between MagiMoney and TechMoney would be driven by supply and demand, which in turn is driven by the foreign trade balance. When a world exports more than it imports, then there will be a surplus of foreign currency in that world, so its price will go down. But when a country imports more than it exports, then foreign currency on the market will become less and the price for foreign currency will go up.
Note that with just two worlds, this is a self-balancing system. Cheaper foreign currency encourages imports and more expensive foreign currency encourages exports. So the balance of trade is going to reach an equilibrium.